Abstract mathematician Eugenia Cheng is the author of a new book about how logic can help us agree—or at least disagree more helpfully. She walks Ira through the fallacies, axioms, and even emotions that can inform our arguments. She tackles everything from how our stance on false positives can guide our opinions about sexual assault accusations and social service policies, to how we spar over privilege, to the economic fights about raising minimum wages.

Cheng says, perhaps counterintuitively, two arguments that disagree can still both be logically sound. And why sometimes, even mathematicians rally emotion to polish off a tricky proof.

Interview Highlights

On finding the logic in a situation.Eugenia Cheng: We can find the logic inside situations even if the whole situation isn’t completely logical. The discipline of mathematics begins with abstraction, where you shave away some of the details that aren’t completely relevant to the situation to find out what’s really going on at the root. And when you’ve got to the root, then you look at the logic. You find that a lot of different situations become the same at the root. And that gives you a way of connecting different situations and tapping into some understanding that you’ve achieved elsewhere and making connections between things.

On starting with the same definitions.EC: [What often happens is that people] simply use the wrong definitions right at the start. This is one of the key ways in which we might think that people are using the wrong facts. There was a survey of undergraduate male students that showed that many of them said that they would have non consensual sex if they thought they could get away with it, but they would definitely not commit rape. Which means they’ve got the wrong definition of rape. That’s something that mathematicians are very careful about: Agreeing on terms right at the beginning.

On using logic if you’re not a mathematician.EC: I think that logic does not require math, but logic is, in a way, part of math. I am a mathematician. And from my point of view, because I am so deeply steeped in the discipline of mathematics, I find that mathematics helps me with logic in the real world. I’m not saying that you have to be a mathematician in order to use logic. And that’s a good thing, because I think everybody can use logic whether they’re mathematician or not.

On if emotion and logic are inherently exclusive.EC: Your emotions are always valid. I think that we have been led into a false dichotomy between emotions and logic. And some people—I’m afraid it’s often men—take the position that if you are emotional, then it somehow proves that you’re not logical, and that’s not the case. I know that I’m a very emotional person, and also, that I’m a very logical person.

They aren’t working against each other. So you don’t actually have to suppress all your emotions in order to be logical. And if anyone tells you that you’re being too emotional and that that means you’re not being logical, that just isn’t true.

Yes, it is very difficult to find the logic in a situation when you’re feeling very strong emotions. And that’s where the mathematical discipline of abstraction really helps me because it means that I can look at a chain of logic and separate that out from the emotion. So I’m not saying I shouldn’t feel emotions, but I put the emotions in a different compartment and think about the chain of logical steps. And this helps me to understand another person’s point of view, even if I disagree with them very viscerally. If I abstract from that and look at their chain of logic, step by step, then I don’t have to feel what they are feeling but I can see the logic that they are using.

On using patience to find logic.EC: Logic is a slow process and it doesn’t happen in the format that is very popular in the current modern world, which consists of a mic drop or a one-liner. Logic takes longer than that. And I wish that we could all take time to slow down our arguments so that we have time to explore the logic.

Segment Guests

Eugenia Cheng

Dr. Eugenia Cheng is Scientist in Residence at the School of the Art Institute of Chicago and author of several books, including Beyond Infinity: An Expedition to the Outer Limits of Mathematics (Basic Books, 2017) and The Art of Logic in an Illogical World (Basic Books, 2018).

Segment Transcript

IRA FLATOW: Ever have an argument with someone you thought was being completely illogical? Maybe it was about what to have for dinner, or something simple like that. Or maybe a little more difficult, about climate change or White House policy.

Whatever the topic, you know it is frustrating, right? So offering up your own side of the story, you feel like you’re getting nowhere, nothing’s happening, nothing rational. Maybe they’re even taking the same point you’re making and using them to support their own side.

Well, my next guest says the hardest part is that we both may employ perfectly sound logic only to come to different conclusions. So to cut through the middle of that and maybe finally pin down a good meal plan or a good climate change response, maybe math can help. And our question for you today, listeners today, is, has anyone’s logic ever changed your mind about something? Did they convince you? Was your mind changed?

We want to know your story. Give us a call, 844-724-8255. That’s 844-724-8255. Or you can tweet us @scifri.

And here to take us through her logic is Eugenia Cheng, mathematician, scientist in residence at the School of the Art Institute of Chicago. Her newest book is The Art of Logic in an Illogical World. And we have an excerpt up at sciencefriday.com/logic. Welcome back, Eugenia.

EUGENIA CHENG: Hi, thanks for having me back.

IRA FLATOW: How’s the piano playing going?

EUGENIA CHENG: Oh, it’s going very well. Thank you very much.

IRA FLATOW: We loved to have you on that.

EUGENIA CHENG: Thanks. That was really fun.

IRA FLATOW: It was fun. You know, you looked around and thought, we all need a better understanding of how logic works. Well, what on earth made you think of that?

EUGENIA CHENG: Well, I have to give credit to my students at the School of the Art Institute of Chicago, because I learned so much from them. And when teaching goes well, it’s a two-way process. And my students have pushed me to think about issues of social justice because they care about those questions so much. And if you can find what your students care about and tap into that, then you can teach them so much better through things that are already motivating to them.

And so I gradually found more and more ways to think about mathematics by asking questions about social justice. And then the reverse happened as well, where I understood a lot more about these questions by using a mathematical framework. My students were so helpful in doing that. I learned so much and thought, well, I really should share this with the rest of the world. And that’s how this book came about.

IRA FLATOW: Well, the 800-pound gorilla in the room, at least for me, is– because we talk we argue all the time here– what is the point of arguing logically if people are bringing emotions to the table and don’t really care about the facts? And now, we can’t even agree on what the facts are. Don’t you need to agree on something to be able to argue logically?

EUGENIA CHENG: It’s a very good question, and it can sometimes seem very futile at the moment, as if maybe we’re just doomed to take sides. And the trouble is that if we think we’re doomed to take sides, then we will be doomed to take sides. It’s a very self-fulfilling prophecy.

Whereas, I have found ways to understand what other people are thinking from their point of view and to find ways of bridging the gap to show that we’re not actually on completely different planets. It’s not black and white. It’s usually that there’s some kind of gray area in between. And if I can find that we’re both on the same gray area, just in different places, then it’s much easier to think about sliding around it than to think about flipping an entire coin onto the other side.

IRA FLATOW: This is Science Friday from WNYC Studios. I’m Ira Flatow, talking with Dr. Eugenia Chang, author of The Art of Logic in an Illogical World. So one of the first things you acknowledge is that the real world isn’t actually logical. But we get a lot of use– but we can get a lot of useful work done in the abstract world.

DR. EUGENIA CHENG: Yes, we can find the logic inside situations, even if the whole situation isn’t completely logical. And the discipline of mathematics begins with abstraction, where you shave away some of the details that aren’t completely relevant to the situation to find out what’s really going on at root. And when you’ve got to the root, it’s there that you look at the logic. And when you shave away those details, you find that a lot of different situations become the same at root. And that gives you a way of connecting different situations and tapping into some understanding that you’ve achieved elsewhere and making connections between things.

IRA FLATOW: But I think that that shaving away process is probably one of the most difficult parts.

DR. EUGENIA CHENG: It might be quite difficult, and it’s one of the things that is off-putting about mathematics. People often say to me, I was fine with the math until the numbers became letters. And somehow, that’s a scary step. But actually numbers are already abstractions. They’re already things where we’ve shaved away the details that two cookies and two apples– well, if you ignore the fact that they’re cookies and apples, then you get the concept of two.

And we can do this with arguments in real life as well, where we can shave away details and discover that arguments about, say, voter suppression have something in common with arguments about cancer screening and arguments about accusations of sexual harassment, for example. Because they’re all about whether we care about false positives more or false negatives more. And then, we can see that those arguments come up in questions about social services, and in questions about who should go to university. And those questions are all over the place.

IRA FLATOW: What are some of the most common ways that you see people being illogical when they argue?

DR. EUGENIA CHENG: Well, one of the basic ones is when they simply use the wrong definitions right at the start. And this is one of the key ways in which we might think that people are using the wrong facts. And so, for example, I’ve seen arguments on the internet. Goodness me, there are some bad arguments on the internet. Never!

But I’ve seen arguments on the internet where people say things like they believe that all immigrants are illegal. And that is not correct. Or they support the Affordable Care Act but not Obamacare. And they’ve got the wrong definition of Obamacare in that case. Or there was a study– a survey of undergraduate male students that showed that many of them said that they would have non-consensual sex if they thought they could get away with it, but they would definitely not commit rape, which means they’ve got the wrong definition of rape.

IRA FLATOW: So we have to agree on terms.

DR. EUGENIA CHENG: We do. And that’s something that mathematicians are very careful about, agreeing on terms right at the beginning. So then, another very common error that people make is when they make a false equivalence between one argument and another argument.

And this often happens in what’s called a straw man argument. But I prefer to call it straw person, because I don’t like gendering things unnecessarily. But if you make an argument that seems quite reasonable to you, sometimes somebody will replace it with a much more extreme argument and believe that that’s what you were trying to say. And then they’ll knock it down, because the extreme argument is much easier to knock down. And that can be very aggravating, because they’re essentially painting you in a very bad light and then taking a moral high ground.

IRA FLATOW: I know that argument. Well, talking with Dr. Eugenia Cheng, author of The Art of Logic in an Illogical World. Join us at 844-724-8255, 844-SCI-TALK. You can also tweet us @scifri. We’re going to take a quick break and come back and talk with Dr. Chang some more, so stay with us.

This is Science Friday. I’m Ira Flatow. In a time of contentious news in divisive political races, can we have better conversations about the issues that get us most emotional? Talking about how a better understanding of logic– logic– can help us talk and listen and maybe start to bridge some of those divides, or at least understand why we’re disagreeing so passionately.

Our guest is Dr. Eugenia Cheng, a mathematician, a Scientist in Residence at the School of the Art Institute of Chicago. Her newest book is The Art of Logic in an Illogical World. And we want to hear from you. Has anyone’s logic ever changed your mind about something? Some listeners are phoning in. And first one, Alex, is from Chicago. Hey, a neighbor.

ALEX: Hi, there.

IRA FLATOW: Hi, there. Go ahead.

ALEX: So I had previously identified myself as an aspiring feminist. And I had done so because my logic was that I really couldn’t fully empathize with some of the issues that women face as a male. And I was met with that logic from a coworker of mine who had said, you know, Alex, as a black male, you might be doing things that contribute to white supremacist systems, that you might do so knowingly or unknowingly. But that doesn’t necessarily make you no longer an African-American male.

In the same way, women or being a man who contributes misogynistic systems unknowingly wouldn’t necessarily make me unable to identify as a feminist. It would just mean that I have to do more to aspire to contribute to feminist systems. And so being able to see the fact that I can still be a male with male experience while existing as a feminists and identifying myself in such a way really changed my mind and the way I thought about that.

IRA FLATOW: Wow. That’s great. Thanks for calling.

DR. EUGENIA CHENG: That is great. That’s very heartening.

IRA FLATOW: Is part of the problem that we don’t see how others see us?

DR. EUGENIA CHENG: I think that is part of the problem. But I think a big problem is that we often spend too long trying to defend our own points of view instead of trying to understand somebody else’s points of view. And we often focus on trying to change someone else’s mind instead of trying to see their points of view and see what its validity is.

And one thing I found very interesting in the caller’s experience is that it was a great example of using some abstract structure to tap into the person’s own experiences. So, in a way, I think the logic was used very cleverly, using emotions as well, to say let’s find an analogous situation that this person will understand from their own experiences and then hope that they can translate that to somebody else’s analogous experience.

IRA FLATOW: I have a tweet from James who says, “To me, logical discussions don’t require math. It requires empathy. Empathy is caring enough to respect another view and listen to it. Is that emotional? I don’t know.”

DR. EUGENIA CHENG: I agree. I think that logic does not require math. But logic is, in a way, part of math. And these things have porous boundaries.

I mean, we’ve constructed a lot of artificial boundaries between subjects for bureaucratic purposes. But I am a mathematician, and from my point of view, because I am so deeply steeped in the discipline of mathematics, I find that the full discipline of mathematics helps me with logic in the real world. I’m not saying that you have to be a mathematician in order to use logic. And that’s a good thing, because I think everybody can use logic whether they’re a mathematician or not.

IRA FLATOW: Let’s talk about some current events. Let’s turn now to the hottest topic in current events. And that is, voters in Congress are divided over whether to believe accusations of sexual assault against Judge Brett Kavanaugh and whether this should affect his confirmation to the Supreme Court. And you write that, in terms of how people respond to stories about sexual assault, there are a number of different logical process at play. Tell me what you mean by that.

DR. EUGENIA CHENG: Well, there are a whole lot of different parts of this argument that we can analyze, starting with what we mentioned before about wrong definitions. Because a lot of people are clinging to the principle of the presumption of innocence. And the thing about the presumption of innocence, I think, is that it’s technically for court cases and criminal trials. And it’s something that gets bandied around as a general principle, but it’s, really, a very specific principle.

And, similarly, the idea of freedom of speech is often bandied around as a general principle. When, in fact, that’s just about whether you can be imprisoned for saying certain things. So going on from the presumption of innocence, there is a paradox. Because some people say, well, we have to presume someone innocent until they’re proved guilty.

And the thing is that if you have an accuser and an accused person– and let’s just say the accuser is a woman, and the accused person a man– then what about the fact that we should also presume the woman to be innocent until she is proved guilty of a false accusation? And so then, we have a paradox, because we can’t actually hold the man and the woman both innocent at the same time, because that causes a contradiction. And mathematics and philosophy have studied paradoxes for a long time and studied ways to resolve paradoxes by shifting levels or finding that things are at a different level.

And the thing is, I think, that we can start with the axiom of presuming the accused person to be innocent. And then we can look at what world that creates for us. And what mathematics does all the time as it starts with some basic assumptions and builds a logical world around that. And then we look at that world, and go, what do I think of this world?

And so we can look at the world that we’ve created by presuming accused people to be innocent until proved guilty. And I think what we should see is that we’ve created a world in which people know that they can get away with sexual assault. Because there is almost never actual evidence that can prove that it happened.

And if they know that they can get away with it, that means that they will do it more. And so we’ve ended up in a culture where sexual assault can happen, and does happen, and happens all over the place because people know that they can get away with it.

So then the question is do we care about that? And that becomes a question of false positives and false negatives. So do we care that so many people are getting away with sexual assault? Or are we more afraid of what some people are worried about, which is that there will be false accusations, and innocent men will have their lives destroyed by false accusation.

So having a false accusation would be when somebody is innocent but is accused of being guilty. And the other way is when someone is guilty but isn’t proved to be guilty and, so, is let off. And that’s really about whether you are prioritizing the plight of victims of sexual assault or the potential innocence of men.

And we have to have a discussion about that and about the fact that we are somehow prioritizing the men who might be accused. We’re prioritizing them over the thousands and millions of women who are suffering sexual assault all the time. And so the question is, are we going to do anything about that?

So the next problem is a question of black and white thinking. And this happens a lot where people get into arguments in extremes. And so many of our arguments, at the moment, turn divisive, because we take extreme opposite positions instead of something more moderate in between. And so there is an argument around saying something like, you’re making everyone into a rapist. Or what one that Professor Langbert said on his blog, that we are characterizing just spin-the-bottle crimes as rape.

And that’s taking a very black and white view of thinking that we are calling all forms of sexual assaults and harassment rape, which is not what is happening. There is a scale. And there are things that are as bad as rape, and there are other things that are bad but perhaps not as bad as rape.

Which takes us to the issue of false equivalences and straw person arguments, where you can take that false equivalence and then you can declare, well, you’re saying everyone is a rapist, which is not what people are saying. Or some people say, you’re saying any woman should be able to destroy the life of any man just by accusing him of sexual assault, which, again, is a very extreme position that I don’t think that anyone really is taking.

And, finally, there’s the question of null and alternate hypotheses, which comes up in scientific testing a lot, where you have to decide what your default position is. And you say, in the absence of further evidence, I’m going to take this default position. And I’m going to wait till I found enough evidence to change my mind.

And the thing is that in a court case and a trial, it’s very important because we’re going to punish somebody badly for something. So before we do that, we should be very sure that they deserve to be punished. Whereas, in the case of a Supreme Court appointment, then what we’re doing is we’re giving them something very good and important. And so we should be very sure that they deserve that very good and important thing.

And, in that case, one might argue that situation could be negated by one bad thing happening. Whereas, in the case of a trial, we should not punish somebody if there is some doubt about whether they should be punished, which means if there is something good to offset our sureness about the fact that they are bad. And those are two– they sound the same. But they’re slightly different situations that have a different null and alternate hypothesis situation.

KARINA: Hi, Ira. Thanks so much for taking my call. My question is– well, I’m wondering how to access my logical arguments when I feel very emotional about the subject that’s being discussed, especially if I feel that I have first-hand experience or background knowledge that supports my logic. But sometimes I feel that my emotions get in the way, and I end up not feeling logical at all.

IRA FLATOW: Good question.

DR. EUGENIA CHENG: Thank you for that question. And I think the first thing that’s important to remember is that your emotions are always valid. And I think that we have been led into a situation of a false dichotomy between emotions and logic. And some people– and I’m afraid it’s often men– take the position that if you are emotional, then it somehow proves that you’re not logical. And that’s not the case.

I know that I’m a very emotional person and also that I’m a very logical person. I’m a successful research mathematician, and my whole profession depends on my use of logic. But I’m also an emotional person.

And those things can go together. They aren’t working against each other. So you don’t actually have to suppress all your emotions in order to be logical.

And if anyone tells you that you’re being too emotional and that that means you’re not being logical, that that just isn’t true. That doesn’t have to be true, anyway. So what I say is that, yes, it is very difficult to find the logic in a situation when you’re feeling very strong emotions.

And that’s where I feel that the mathematical discipline of abstraction really helps me, because it means that I can look at a chain of logic and separate that out from the emotion. So I’m not saying I shouldn’t feel the emotions, but I put the emotions in a kind of different compartment and think about the chain of logical steps.

And this helps me to understand another person’s point of view, even if I disagree with them very, very viscerally and I’m very upset about their position. If I abstract from that and look at their chain of logic step by step, then I don’t have to feel what they are feeling, but I can see the logic that they are using. And those two things can be separate. But then I can actually use the logic to be able to understand their points of view despite my emotional disagreements.

IRA FLATOW: This is Science Friday from WNYC Radio. I’m Ira Flatow talking with Eugenia Chang, author of The Art of Logic in an Illogical World. I think you know that is one of the hardest things to do these days is to just count to 10.

DR. EUGENIA CHENG: Yes. That’s a very important aspect of it, because logic is a slow process, and it doesn’t happen in the format that is very popular in the current modern world which consists of a mic drop, or a one-liner, or some kind of meme that doesn’t have very many words in it, or a 280-character tweet. Logic takes longer than that. And I wish that we could all take time to slow down our arguments so that we have time to explore the logic and build the logic.

IRA FLATOW: So it’s almost like mindfulness. You have to find a way to catch yourself in the moment.

DR. EUGENIA CHENG: Yes, it is a lot like that. And I think that that kind of mindfulness can help us understand the logic inside our own emotions. Because our emotions are all coming from somewhere. There is something that is causing them.

And I honestly believe that when I see somebody who is feeling something emotional about something very differently from me, if I do that mindfulness for them, even if they are unable to do it for themselves, then I can start to understand what their basic belief is from which all of this is stemming. And maybe I can see how that basic belief is different from my basic belief. And that’s why we think different things, not because we our logic is different, but because our starting points are different.

IRA FLATOW: Well, I have about a minute or so left. I want to ask you about something you write a great deal about the logic of privilege relating, especially, to race and gender. Is that an area specially ripe for illogical treatment?

DR. EUGENIA CHENG: Yes, the arguments get very vehement, and some people get very upset about the whole question of privilege and identity politics. And I think it’s important for us to understand everyone’s points of view. And, often, it’s a misunderstanding about what privilege means.

Because, for example, white privilege does not mean that all white people are better off than all non-white people. It means that any white person, if they had all the same characteristics and the exact same life, except that they were black as well, we would expect them to be worse off in society. And so some white people get upset because they say things like, but I grew up poor, and I had to work really hard, so that means that I didn’t have white privilege.

And we need to acknowledge the fact that, indeed, there are white people who are really struggling. And we shouldn’t let our discussion of privilege erase that fact that they’re are white people who are struggling. Because if we erase that fact, then they get angry and feel that they have been disenfranchised and that they have no voice. But we need to understand that if they were black, then they would have an even worse time. And that’s what the logic and the structure of privilege is really about.

IRA FLATOW: Dr. Cheng, thank you for joining us again. As always, it’s a pleasure to have you.

DR. EUGENIA CHENG: Thanks very much.

IRA FLATOW: And good luck with the book. Eugenia Cheng, mathematician, Scientist in Residence at the School of the Art Institute in Chicago. Her newest book is The Art of Logic in an Illogical World. And we have an excerpt up at sciencefriday.com/logic.

One last thing before we go, this week the world lost a Nobel physicist, a pioneering educator, and a popularizer of science, Dr. Leon Lederman. Not only was he a director, Fermilab, he was a tireless fighter for science literacy seeking to reach the public about the importance of science through any means possible.

DR. LEON LEDERMAN: It’s a battle to keep our science programs viable. And that’s one of the problems I think we have. There are obstacles towards the goal we have, and we think it’s an incrucial goal to improve our teachers, improve our education, and get the reasonable number– in fact, the general population, the graduates of our high schools and colleges to be comfortable with science and to be skeptical that you can do everything with science, what you can do and what you can’t do.

This is public science literacy. It’s crucial. And if we have that, then we’ll have better teachers and better students. And maybe we can pull out of this slump.

IRA FLATOW: Nobel Laureate Leon Letterman speaking to us on Science Friday in 2006. He died this week at the age of 96. Condolences to all his friends and family.

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